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ODERACS 2 White Spheres Optical Calibration Report
Culp, Robert D. 1 Gravseth, Ian 2 Wantuch, Todd 3
Gloor, Jason 3
Colorado Center for Astrodynamics Research
Department of Aerospace Engineering Sciences
University of Colorado
Boulder, CO 80309-0431
University Contract No. 1536400
NASA Reseach Grant NAG 9-768
January 9, 1995
1 Professor and Chairman
2 NASA Graduate Student Research Fellow
3 Undergraduate Research Assistant
Contents
1 Introduction
2 Methodology
2.1 Procedure
2.2 Light sources
2.3 Spectrometer
2.4 Phase angle
2.5 Data processing
3 Results and Discussion
3.1 Spheres
3.1.1 Albedo and scattering
4 Conclusions
4.1 Acknowledgements
4.2 References
A ODERACS Project, Spheres and Tasking
A.l ODERACS project
A. 2 Tasking
A.2.1 Task : Flight Spheres Calibration .
B Experimental Setup
B. l Experimental Setup
C Procedure
D Spectrometer
E Light Source
F Object Handling
G Data Processing
G.l Measurements
G.2 Spheres
G.2.1 Phase angle correction
G.2. 2 Range correction
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G.2.3 Estimating albedo 25
G.2.4 Reflectance characteristics 26
G.2.5 Analysis and presentation of the sphere information 30
G.3 Data organization 30
H Flight Sphere Results 31
List of Tables
3.1 CRC values for reflection coefficients
3.2 Scattering and albedo values for white spheres
H.l Scattering and albedo values for white spheres
List of Figures
2.1 Basic experiment setup
2.2 Direct light signal vs wavelength
2.3 Phase angle definition. This is a top view of the sphere-sensor setup (not to
scale)
2.4 Diffuse versus specular signals
3.1 Reflectance of metal mirrors from Allen . . .
B.l Basic experiment setup - side view
B.2 Main light source configuration
B.3 Sensor mount configuration
B.4 Sphere mount
D. l Signal drop off versus off-axis viewing angle .
E. l Light calibration results
G.l Albedo calculation geometry
G. 2 Geometry for specular sphere reflection . . .
H. l 4 inch white flight sphere ID number snl008 .
H.2 4 inch white flight sphere ID number snl012 .
H.3 4 inch white flight sphere ID number snl008 .
H.4 4 inch white flight sphere ID number snl012 .
10
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4
Abstract
This report documents the status of the Orbital Debris Radar Calibration Spheres
(ODERACS) 2 white spheres optical calibration study. The purpose of this study is to
determine the spectral reflectivity and scattering characteristics in the visible wavelength
region for the white spheres that were added to the project in the fall, 1994. Laboratory
measurements were performed upon these objects and an analysis of the resulting data was
conducted. , ,
These measurements are performed by illuminating the objects with a collimated beam
of light and measuring the reflected light versus the phase angle. The phase angle is defined
as the angle between the light source and the sensor, as viewed from the object. By
measuring the reflected signal at the various phase angles, one is able to estimate the
reflectance properties of the object.
The methodology used in taking the measurements and reducing the data are presented.
The results of this study will be used to support the calibration of ground-based optical
instruments used in support of space debris research. Visible measurements will be made
by the GEODDS, NASA and RADOT telescopes.
Chapter 1
Introduction
This report documents the Orbital Debris Radar Calibration Spheres (ODERACS) 2 pre-
flight optical calibration study of the two supplemental white spheres delivered to the Uni-
versity of Colorado in fall, 1994. The paper presents the visible region spectral reflectivity
and scattering characteristics for the ODERACS 2 white spheres. The results of this study
will be used to support the calibration of ground-based optical instruments in support of
space debris detection research.
This report is organized such that the main body contains the essential information
and the appendixes axe used to present a more detailed analysis. The basic setup, data
reduction methods, results, and discussion of the results are presented. More information
on the ODERACS project, the test objects, and the tasking can be found in Appendix A.
1
Chapter 2
Methodology
The setup and procedure for the preflight optical calibration of the spheres were driven by
the need to have both scattering and specular measurements sufficient to find the albedo
over the visible region from 450 nm to 950 nm wavelengths. The statement of work specified
that the measurements were to be taken, to the extent possible, at phase angles from ~0 to
~180 degrees in 5° intervals. The measurements were taken over all possible angles where
the sensor did not interfere with or enter the light beam.
Measurements were taken for the two white spheres. A spectrometer capable of per-
forming efficiently this large number of measurements was used. The spectrometer uses a
fiber-optic cable to attach the sensor to the processing box. This cable allows the sensor
to be moved easily for the many phase angle measurements. Appendix D contains more
information about the spectrometer. Figure 2.1 shows the basic setup for the experiment.
Appendix B contains more sketches and discussion of the apparatus and setup.
2.1 Procedure
Before the flight hardware was tested, the experimental system was aligned and tested. A
rehearsal of the experimental process was conducted to assure correct handling of the flight
hardware. Appendix F contains more information on the handling precautions implemented
for the experiment. . . ,
Once the equipment was ready, the objects were placed in position and measurements
were taken over the entire phase angle range in a timely manner. Measurements of the
direct light signal were made for use in calculation of the albedo. This process was repeated
for each object. After each object was tested, it was returned to its carrying case.
2.2 Light sources
The objects were illuminated with a 1000 Watt Quartz-Halogen light source reflected
through a planar secondary mirror and a high precision parabolic mirror to form a highly
collimated light beam which closely resembles solar conditions. This high energy light source
was needed to provide enough light for the spectrometer to work accurately and efficiently.
The direct signal spectrum for the 1000 W light is shown in Figure 2.2 It can be seen
that at extreme wavelengths the direct signal is not very strong. Due to the nature of the
2
light source and the noise of the spectrometer, the data is not reliable below 450 nm or
above 950 nm.
2.3 Spectrometer
The spectrometer used in this study is a LabSpec Spectrometer, developed by Analytical
Spectral Devices, Inc (ASD) of Boulder, Colorado. The sensor, which feeds the spectrome-
ter, is attached via a fiber-optic cable to the body of the spectrometer. This made it easy
to take measurements at various phase angles.
The spectrometer is run by a PC, which drives the spectrometer and stores the data.
It was found that the drift in the noise of the spectrometer was noticeable for the longer
integration times and weak signals. More information on the spectrometer is found in
Appendix D. Information on how the drift was removed from the data is found in Appendix
G.
2.4 Phase angle
Measurements at many phase angles were taken so that the albedo and the percentage of
light scattered could be estimated. The phase angle 0, as shown in Figure 2.3, is the angle
between the line from the test object to the source and the line from the test object to the
sensor, as viewed from the object. The measured angle 0 and the actual phase angle 2<p
for the light measured are somewhat different because of the limited range to the sensor.
The difference between these two values can be easily calculated from the geometry of the
setup, and is discussed in Appendix G.
Multiple phase angle measurements allow the scattering and specular signals to be de-
coupled and gives one the ability to estimate the total reflected light. This total reflected
light is used to obtain the albedo estimate. There are several definitions for albedo, which
are described in Appendix G. For this work albedo is:
3
( 2 . 1 )
tot al reflected light
Albedo = tota j i nc id e nt light
F or a perfect specffiar reaction, under
a constant intensity from evenly collima.te g ^ ^ I ^ mbertiail surface, one sees a very
For a perfectly diffuse surface, whic sea polished spheres, the reflection
distinct falloff in signal with phase angle. are observed for the
is not perfectly specular. Both specular and scattering
white spheres. part8 of the light signal from a perfectly
The following equations lllustra tivelv These equations represent the ideal
diffuse and perfectly specul* rrf««md, such tha" the source can be treated
diffuse and specular signals viewe Appendix G. Figure 2.4 shows the idealized
^albedo equal to one and with 50 percent
of the light scattered diffusely.
Diffuse
37 rR 2
Specular
E r =
aEr 2
4uR 3
( 2 . 2 )
(2.3)
where
E r =
E =
a =
r =
R —
e =
Reflected flux
Incident flux
Albedo
Radius of sphere
Range to sensor
Phase angle
2.5 Data processing
Due to the large number of ^atdy^ J
the most difficult portion of tiu^ Pr °^ bc d 0 scattering result, were reasonable,
data from the object, to determine if he albedo ^ carrfully.
Then at a later time, the data was P lott ' d “ p “ J aU are idealised equations with
The above equations for the specular and spectrometer and setup, the ratio
the far held approximation. Duetothemia Reid approximation is
of a to r was between 5 mid 25. For the ^ r were used in the albedo
violated. Therefor. modiSed value, for the phm< ’
estimation. This is discussed more in Appen
Umbwt. HO*-." c0
6
Normalized Flux Ratio
Angle
Figure 2.4: Diffuse versus specular signals
7
Chapter 3
Results and Discussion
, . 1t f tbe s tudy and a discussion of those results. The
£££££. * ** «* - - d * u plot * for the ,wo wW,e sp
are located in Appendix H.
3.1 Spheres
The reflectance propertie. were t^abte >«« for
■*— again below -her. the term on the
Stand ^ of the eguatron i. referred to a. the normalmed ..gnal.
(3.1)
E r l l 2
Er 3
(-7
— [sinfl + (x-®)coi«] + (l-7)-f
3ir
E r
E
a
r
R
e
7
£
Reflected flux
Incident flux
Albedo
Radius of sphere
Range to sensor
Phase angle
Percentage of light scattered
Proportion of Sphere Viewed (Due to limited eeneor range)
The albedo Ld ecattering vain., for the
in Tables 3.2. The results for these surfaces appear consistent wi
visual inspection of the surfaces. ^ tQ be the drift of the spectrometer and a
The main errors in reducing th blem is discussed in detail in Appendixes D
small range-to-radius ratio. The ba8 a certa in background noise which changes
and G, but is outlined bereThespecU ^ fluctuate8 the background noise drifts,
with temperature. As the temp of measuremellt8 C an be of the same
This is a problem because the drift rate over
8
Material
Nature of surface
Coefficient
Aluminum, Polished
Specular
0.69
Steel, Polished
Specular
0.55
Chromium
Specular
0.62
Platinum
Specular
0.62
Table 3.1: CRC values for reflection coefficients
magnitude as the actual measurements. In addition to the drift, there is some random noise
in the signal.
Many measurements were made of the drift rate for the spectrometer. This drift rate
can be effectively removed for the wavelengths with the stronger signals. Due to the much
smaller light strength at the extreme wavelengths, the errors at these wavelengths are
expected to be higher.
The low signal-to-noise ratio for the spectrometer under the lighting conditions for this
experiment force the sensor to be placed close to the spheres. This in turn created a low
range to sensor over radius of sphere ratio, j. The equations shown for the expected fluxes
from the diffuse and specular components were derived for a sensor at a large j ratio.
These equations are still reasonable approximations for this study, but modifications to
these equations are being studied.
A low value for this ratio also meant that the range to the specular reflection point was
not constant over the measured phase angles. Correcting for this varying value of the range
was performed for the white spheres. The same range correction was used for the white
spheres as for the specular reflecting spheres in previous ODERACS tests. The correction
is discussed in Appendix G.
Since there is a falloff in signal when the sensor is not pointed directly at the incoming
light and the white spheres reflect a portion of light diffusely, the measured signal will be
somewhat lower than its true value. This error was not compensated for and will reduce
the measured signal an amount depending on the ratio,
Attempts were made to compensate for all the above mentioned errors.
3.1.1 Albedo and scattering
The albedo and scattering values for the white spheres are presented in Table 3.2. These
values represent the average and standard deviation for the best measurements on the
individual spheres. It should be noted that the estimates for the scattering have much
larger errors than the standard deviation indicates.
The albedo values from both estimation methods agree well with each other for the
spheres.
The estimates of the scattering for the spheres were very sensitive to small changes in
the signal and should be treated with caution. The scattering component of the spheres
comes from the off-bore sight directions, which suffer from signal degradation. This will
cause the scattering component to be underestimated. Unfortunately, only one method was
used to estimate the scattering component for the spheres, so there is not a backup method
with which to compare the scattering estimates.
9
Chapter 4
Conclusions
The methodology and results for the ODERACS 2 white spheres preflight optical calibration
study have been presented. The calculated albedo and gamma coefficients for both white
spheres are consistent between tests.
4.1 Acknowledgements
The authors would like to acknowledge two people for their contributions to this study.
Dr. William McClintock of the Laboratory for Atmospheric and Space Physics (LASP) for
the use of his laboratory, equipment, and time, and also Glen Cress for his role as liaison
between CU and JSC.
4.2 References
Allen, C.W. Astrophvsical Quantities. Third Ed., The Athlone Press, London, 1973, pg.
108.
CRC Handbook of Chemistry and Physics . 62nd Ed, CRC press, Boca Raton, 1981, pg
E-386.
Egan, W.G. and T. Hilgeman. Applied Optics , Vol. 15, No. 7, July 1976, pp. 1845-1849.
Egan, W.G. and T. Hilgeman. Applied Optics, Vol. 16, No. 11, November 1977, pp.
2861-2864.
Hapke, B. and E. Wells, ’’Bidirectional Reflectance Spectroscopy I - Theory”, Journal of
Geophysical Research, Vol 86, No B4, April 10, 1981, pp 3039-3054.
Hapke, B. and E. Wells, ’’Bidirectional Reflectance Spectroscopy II - Experiments and
Observations”, Journal of Geophysical Research, Vol 86, No B4, April 10, 1981, pp
3055-3060.
Lompado, A., B.W. Murray, J.S. Wollam, and J.F. Meroth, “Characterization of optical
baffle materials.” SPIE Proceedings Series: Scatter from Optical Components , Vol.
1165, 1989, pp. 212-226. Proceedings of a conference, 8-10 August, 1989, San Diego,
CA.
11
Madler, R.A., R.D. Culp, and T.D. Maday. “ODERACS II pre-flight optical calibre-
tion.” To be preseated as the 1993 SPIE Aerospace and Remote Sensing Conference,
Orlando, Florida, 12-16 April, 1993, Paper 1951-06.
Marx, E. and T.V. Vorburger. “Light scattered by random surfaces and roughness -
mination.” SPIE Proceedings Series: Scatter from Optical Components, Vol. 1165,
1989, pp. 72-86. Proceedings of a conference, 8-10 August, 1989, San Diego, CA.
Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vertterling^ Numerical Recip .es:
The Art of Scientific Computing Cambridge University Press, Cambridge, 1986.
Rambauske, W.R. and R.R. Gruenzel. J. Am. Opt. Sac., Vol. 55, No. 3, March 1965, pp.
315-318.
Stover, J.C. Optical Scattering: Measurements and Analysis. McGraw-Hill, N.Y., 1990.
Veverka, J. “Photometry of Satellite Surfaces,” in Planetary Satellites, ed. Joseph Burns,
University of Arizona Press, Tucson, 1977.
12
Appendix A
ODERACS Project, Spheres and
Tasking
A.l ODERACS project
This report is an important part of the NASA Orbital Debris Radar Calibration Spheres
(ODERACS) 2 project. The overall goal of this project is to provide reference targets for
the calibration of both radar and optical sensors for small orbital debris objects.
The purpose of this study is to determine the spectral reflectivity and scattering charac-
teristics in the visible wavelength region for various objects. These measurements are used
to determine an approximate albedo and phase angle scattering as a function of wavelength
for the spheres.
ODERACS is a project of the Solar System Exploration Division at JSC, and is a
joint effort between JSC, GSFC, NCSU, DoD, Phillips Lab, the USAF, the University of
Colorado, and others.
A. 2 Tasking
The statement of work for the preflight optical calibration is summarized below.
A. 2.1 Task : Flight Spheres Calibration
Using unpolarized light, the absolute visible spectral reflectivity and scattering characteris-
tics of all flight spheres are to be determined between 4500 and 9500 Angstroms. The data
is to be sufficient to obtain albedo, and to generate phase angle scattering plots as a function
of wavelength. Spectral measurements are to be made within bands of 800 Angstroms, or
less, and at intervals of 500 Angstroms, or less. Scattering measurements are to be made
at phase angles from near-zero degrees to near 180 at 5 degree or smaller steps. Plots and
tabular listing of the spectral reflectivity as a function of scattering angle are to be prepared
for each sphere.
If it may be reasonably assumed that the spheres are uniform, there will be not need
to ex amin e more than one portion of the sphere. However, since this is not likely to be the
case, the spheres should be measured at five individual orientations.
13
Appendix B
Experimental Setup
B.l Experimental Setup
The purpose of this experimental setup was to simulate as closely as possible the solar radi-
ation incident upon an object in orbit about the Earth in order to estimate the reflectance
characteristics for several objects. The light that was used had to be collimated (parallel
light rays) and controlled to fall only on the object that the spectrometer was measuring.
All other light was blocked out by means of baffles, see sketches. The primary light source
used was a 1000-watt light bulb with a constant power supply, see Appendix E.
Figure 1 details the basic equipment setup used in this experiment. The sphere mount
and parabolic mirror were placed at opposite ends of the table, primarily because of space
constraints. The light source was placed at a distance of 90 inches from the center of the
parabolic mirror, the focal length. Figure 2 details the 4 inch flat mirror was used to reflect
the initial light beam into the parabolic mirror. The distance from the bulb to the flat
mirror and from the flat mirror to the parabolic mirror was 13 and 77 inches respectively.
Several measures were taken to absorb or block extraneous light. The experiment was
conducted in a black painted room and all light from outside was sealed out. The baffles
used were fitted carefully together and covered over the top with black felt, forming a large
black box in which the light source was placed. This allowed only a simple beam of light to
emerge past the baffles. However, the parabolic mirror was not contained in this enclosed
region because it had to be placed some seven feet away at one end of the table. This was
the primary source of extraneous light. Another source of unwanted light was from the
mount on which the objects were placed. Even though the collimated beam illuminated
only the target, secondary reflectance was produced from the light reflected down from the
target and onto the mount.
Figure 3 details the sphere mount used in this testing. The mount itself had a built in
angular protractor which made angle measurements easy and accurate. The mount stood
11 in. high and was covered with black felt. Additionally, an extended platform (arm) was
connected to the side of the mount which was used to attach the spectrometer’s sensor.
14
Figure B.l: Basic experiment setup - side view
The spheres were placed onto a ring to prevent them from rolling off the horizontal
surface. Figure 4 shows a diagram of the sphere mount. As a result a very small portion of
each sphere dipped below the rim of the ring and was hidden from the light beam. It was
felt that this was a logical step so that the safety of the spheres could be insured.
15
16
-I 1 1-
Figure B.3: Sensor mount configuration
Figure B.4: Sphere mount
17
appendix C
procedure
tit was set up and
Each day before ^tTappene^ M '««*• sUnch
tmovS tm^storaje in the
toom were -««£*£» were then «“ '*'“ f£t
measurements. possible without letting dr tft cou i<i be measure . Y
to zero degrees as was p that the spectrometer q{ the sensor could
Several dark reading, -ere ak ^ ^ 16 5 ^eea j^ f „„ m ea.««d, a
icaily, 7 V d „wn to fraction, of a degree- After
be measured accura y , signal was taken. ^turned to the carrying case.
meu wrrkrl«nrer.n»-
* omx pli,£oim " 1
18
Appendix D
Spectrometer
The spectrometer used for this laboratory experiment was a LabSpec, designed and built by
Analytical Spectral Devices, Inc. (ASD) of Boulder. It has 512 channels for data sampling,
utilizing a plasma coupled photodiode array, with spectral range of 347.7 to 1056.6 nm. It
has integration times ranging from 17 milliseconds up to 9 minutes. The spectrometer’s
noise can be removed either manually or automatically. The sensor utilizes fiber optics, and
it has a two meter cable so that the sensor can be easily moved. The sensor itself consists
of a fiber optics bundle that is roughly .6 mm in diameter at its terminus.
Although on the whole the spectrometer was well suited for this experiment, it also
had a significant amount of drift in its dark signal over time, partially due to temperature
fluctuations within the room. While this drift would be relatively insignificant for mea-
surements being taken under sunlight, with this laboratory experiment the amount of noise
in the signal was significant, especially when viewing the test article with long integration
times and small reflected signals.
Since the signal had a significant drift in it, the drift was subtracted from the signal
by assuming that the drift between two successive dark measurements was approximately
linear. This provides an acceptable estimation of the signal for the wavelengths with a
strong signal, but causes the extreme wavelength calculations to be unreliable. For this
reason the extreme wavelengths have not been used in the calculations.
This spectrometer comes with a laser pointing device so that the sensor is easily pointed
at the specular point. The laser pointing device is necessary to correctly align the bore sight
of the sensor. This is necessary due to the signal falloff at angles offset from the bore sight.
This signal falloff as a function of off-axis angle is shown in Figure D.l. For the specular
spheres analyzed in previous experiments this does not introduce significant errors due to
the specular nature of these objects. However, for the more diffuse objects a significant
signal comes from the off-bore sight directions. For this reason, these objects were viewed
from as great a range as possible to reduce the maximum off-axis angle. Nevertheless, the
estimated albedos of the diffuse objects are most likely underestimated in this experiment.
19
Appendix E
Light Source
All the objects were illuminated with a 1000 Watt Quartz-Halogen light source reflected
through a high precision parabolic mirror to form a highly collimated light beam which
closely resembles actual solar conditions. The lamp was a commercial GE type PEL 1000-
watt lamp having a tungsten coiled-coil filament enclosed in a small quartz envelope. The
focal length of the Parabolic mirror was 90 inches. The resulting beam has divergences
ranging between 0.5 and 0.7 degrees, based on and limited to the accuracy of the actual beam
projection dimension measurements. This compares well with the actual sun conditions of
0.53 degrees. See Appendix B where Figures B.l and B.2 show the setup of the light source.
This lamp was powered by a constant power supply source which provides a constant 8
amperes of current to the light. Figure 2.2 shows the raw data from a direct measurement of
the light. This signal shows the spectrum of the light source as sensed by the spectrometer.
This can be compared with the calibration of another light bulb of the same model performed
by Optronic Laboratories, Inc. The spectral irradiance is given in a * a distance
of 50 cm when the light is operated at 8.0 amperes. 1 The differences between Figs. 2.2
and E.l are most likely due to spectrometer characteristics and some varaince in the signal
produced by each bulb.
1 Letter from Optronic Lab dated 19 March, 198S
21
Appendix F
Object Handling
All tested objects were handled with equal caution. The shipping cases which contained the
spheres were fastened with padlocks. At the end of each day the spheres were placed in the
controlled inventory room at the Laboratory for Atmospheric and Space Physics (LASP)
and secured.
Once an object was taken out of the case it was placed immediately upon the dust-free
rubber ring-mount on the main mount (see Appendix B) which was surrounded by high-
density foam similar to the foam inside the spheres* cases. Special lint-free gloves over latex
gloves were used when handling the objects.
All of these handling procedures were rehearsed before the spheres were removed from
the protective cases. As a result of the rehearsal, the experiments proceeded smoothly
without incident.
23
Appendix G
Data Processing
This section deals with the methodology used in the data processing and handling. The
following is a listing of the basic steps which were taken in analyzing the reflectance data^
Several programs were written to analyze the data for each different type of object being
tested. The following sections discuss the analysis of the spheres.
Data Processing Procedure
1. Take measurements.
2. Quick analysis of results.
3. Transfer data to floppy disks for more analysis and plotting.
4. Analyze the information.
G.l Measurements
As mentioned before, the spectrometer stores information for each of is 512 channels at each
phase angle. Measurements are taken at 5 degree intervals for all possible phase angles.
These measurements are stored, the data is analyzed, and then the files are transferred to
floppy disk for storage. This was very straightforward in the PC operating environment.
G.2 Spheres
This section discusses the data analysis for the spheres. The albedo for the spheres was
calculated using two different methods. The scattering was estimated with only one method^
The following paragraphs discuss some of the corrections made to the sphere data and the
methods of estimating the reflectance properties. ,. . . .
The phase angle and range corrections discussed below come about because limitations
with the P setup do not allow the spheres to be viewed from a far distance. This means
,h.i the spheres do pot act a. a point source and trill not follow the simplified reflectance
equations perfectly. The corrections attempt to rectify this problem.
24
G.2.1 Phase angle correction
The geometry of the phase angle problem is given in Figure 2.3. Though this drawing is
not to scale, it can be seen that the actual phase angle of the light off the sphere is 2<f>
instead of B. Therefore the measured values of the phase angle are corrected to take this
into account. The following equations give the relationships between the sides and angles
in Figure 2.3. From the law of cosines
b 2 — t 2 + R 2 - 2rfZcos (9 — 4>) (G.l)
From the sum of angles in a triangle.
7 = 2 <t>-B (G.2)
From the law of sines and previous equation.
6 = 2<p - sin -1 (G.3)
it
where
r = Radius of sphere
R = Range to sensor
b = Modified range to sensor
9 = Measured Phase angle
2 <p = Modified Phase angle
7 = Offset angle
G.2.2 Range correction
The short range to the sensor causes other problems because the ratio £ is not close to 1.
This means that the sphere cannot be considered a point and the distance to the reflection
point changes with changing phase angle. The range from the reflecting surface to the
sensor is given as b in the phase angle correction section.
There is a severe problem at this point when talking about the diffuse case. The diffuse
scattering of a sphere actually would have many ranges, it is not a trivial problem. This
problem is not taken into account in our formulation, but is another source of possible error.
This changing range is important to take into account when fitting the data to the equation
for a combination of specular and diffuse signals. The range correction is not needed for
the albedo estimation which is obtained by direct integration of the reflected light.
G.2. 3 Estimating albedo
The most crucial task of this experiment was to find the reflectance characteristics of the
spheres so that the albedos could be found. The Albedo, sometimes referred to as the Bond
Albedo (Ab), is a measure of the reflectivity of a surface, that is, the percentage of light that
the surface reflects in all directions. 1 Another type of albedo is the Geometric Albedo,
^Trom Hartmann, Moons and Planets
25
total reflected light
Albedo - tQta j incident light
Reflected Light = f E r {9) • A(9)d8
Jo
A(0) = Area at this phase angle
A(8) = 2 ir ■ R 2 (cos9i - cosflj) for 9 2 > 9 X
Ad
A(9) = 4ir • R 2 • sin 9 avt • sin — for A0 > 0
Incident Light = irr 2 ■ E
the percentage of Ugh. reflected a. sero phaee angle, compared to a perfect
the percen g 6 2 TH B n d Albedo definition for albedo of Eqn. 4 is
surface with equal projected ar • integrating the measurements over
used io thi. study. The total reflected Ugh . eft mnd ^n. g g t0 obt ain the
the measured phase angles, Eqns. 5-8. The total renecieu ug
albedo estimate, Eqn. 9.
(G.4)
(G.5)
(G.6)
(G.7)
(G.8)
(G.9)
These equation, are implemented by numerically integrating the refl
problem.
G 2 4 Reflectance characteristics
The multiple phase angi. me*—
decoupled for the spheres. For a perfectly specular ““ d u h , coming 0 g a
conditions, «“ ""“T *(.““‘^1^ L. surface one see. a very distinct faUolf
zzsxa ®nth ssj - — * — - - the
spheres in this experiment. . , . • A what tvne of reflectance
The surface of a. either
characteristic, .t wffl have. The „ rf J such as a mirror almost all the
specular or scattered hg**- J e/hand a Lambertian surface reflects (scatters) the
light reflected is specular. On the other han , assumed that the spheres will
light equally in all direction. In <*• “P— - reflections'
reflect with a linear “ ^he ideal reflected flux of a diffuse and specular sphere
v dl.T«« i£- equation, com. out of note, from Dr. lohn Lambert of
Rockwell. 3
' IJS^sSSSSfSSfS 1 ^ »— • la... Colo. Sp„, CO
26
I
where
Figure G.l: Albedo calculation geometry
Diffuse sphere
2aEr 2
E r = gy oa 0+{*-9) cos 9 ]
Specular sphere
aEr 2
E r =
4 R 2
E r = Reflected flux
E = Incident flux
a = Albedo
r = Radius of sphere
R = Range to sensor
9 = Phase angle
(G.10)
(GU)
The equations for the specular and diffuse conditions assume that the object is either a
perfect diffuse or specular object. This is not a realistic condition because to some extent
27
objects will have a combination of diffuse and specular reflection. Equation G .12 combines
the diffuse and specular equations using the variable 7 to represent the percentage of light
scattered diffusely. This assumes that the reflected light can be represented by the linear
combination of these two idealized signals. The term on the left hand side of the equation
is referred to as the normalized flux ratio for the spheres. The normalized flux ratio for
an idealized sphere with an albedo equal to one which scatters 50 percent of the light is
shown in Fig. 2.4. This figure shows the diffuse, specular and total signals reflected.
= ^~t sin ^ + ( ,r_5 ) C08 ^ + ( 1 " 7 )' \ ( G - 12 )
where
7 s Percentage of light scattered
£ = Factor for not viewing
the entire sphere
Values for the percentage of specular and diffusely reflected light may be obtained by
estimating the values of 7 and a in Eqn 12. This technique is able to estimate the albedo
and scattering even with a limited set of data, though sometimes the fit does not follow the
data trends accurately. In Eqn. 12, the reflected light signal depends non-linearly upon the
phase angle and linearly on the albedo. Numerical non-linear fitting techniques are used to
estimate the values for the albedo and percent of light scattered, 7. 4
Equations G .10 - G .12 use the far field approximation - that the object is far enough
away to treat it as a point source. Due to limitations within this experiment, this approxi-
mation was violated. The equation for the specular reflection off a sphere is easily modified
for the close range measurements. The derivation without the far field approximation was
done by modifying notes from Dr. John Lambert of Rockwell Int., Colo. Spr., CO. Figure
G.2 shows the geometry for the specular case. The incoming light has an intensity E which
is incident upon a small area defined as The reflected light has an intensity E r
which is even over the area *’( r jliH - . The total reflected light equals the total incident
light times the albedo. This yields:
Specular E, = (4t , + r i j < G13 >
It can be seen that in the limi t as b approaches zero, and for values of b much greater than
r, this equation converges to the previous specular equation. This derivation is obtained
by assuming area of the incoming light can be approximated by the the area given above.
This derivation is not strictly correct, but should model the range correction much better
than the idealized equations.
The derivation of the equations for the spheres was not integrable in the modified form.
The equation can be numerically integrated for specific sphere sizes and ranges to the sensor.
Due to the non-integrability of the diffuse equations, the spheres were evaluated using this
range correction.
‘Numerical Recipe*
28
Figure G.2: Geometry for specular sphere reflection
29
G.2.5 Analysis and presentation of the sphere information
The information from this study is presented in both tabular and graphic format to illustrate
the properties of each sphere. A PC FORTRAN program has been written to analyze the
data after the measurements are taken to verify that the data looks reasonable. The basic
program flow is outlined below.
Sphere albedo analysis
1. Read in the light signal.
2. Read in the measurements at different phase angles.
3. Correct for the spectrometer drift.
4. Calculate the albedo by the direct integration method.
5. Correct for the phase angle and range to sphere.
6. Calculate the albedo and scattering by non-linear parameter fit.
7. Utilize a UNIX platform for printing and plotting results.
G.3 Data organization
The data for this project was organized according to the naming convention used by Lock-
heed. Additional information regarding the test number was also included in the data file
names. An example of the naming convention used is: SN1008T1.035, which indicates the
data file in question is the 35th data file (.035) of the first test (Tl) on the SN1008 white
sphere.
i
30
Appendix H
Flight Sphere Results
The plots generated from the flight sphere data are presented in this appendix. Table H.l
summarizes which figures correspond to which flight sphere. Five tests were performed on
each sphere. The first two figures show the albedo-scattering plots for each white sphere,
while the second two figures show the best fit and raw data for each of the two spheres.
Sphere
ID
Size of
Sphere
Type of
Sphere
Albedo
Deviation
Scattering
Deviation
SN1008
4
White
.7794
.041
.8168
.014
SN1012
4
White
.7654
.031
.8216
.054
Table H.l: Scattering and albedo values for white spheres
31
signal and fit
snl 008 t 2 fit
Figure H.3: 4 inch white flight sphere ID number sn!008
34
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